MinT - Best Known (54−31, 54, s)-Nets in Base 128

Best Known (54−31, 54, s)-Nets in Base 128

(54−31, 54, 408)-Net over F₁₂₈ — Constructive and digital

Digital (23, 54, 408)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (3, 18, 192)-net over F₁₂₈, using
  - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
      - a function field by SÃ©mirat [i]
2. digital (5, 36, 216)-net over F₁₂₈, using
  - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
      - a function field by SÃ©mirat [i]

(54−31, 54, 514)-Net in Base 128 — Constructive

(23, 54, 514)-net in base 128, using

2 times m-reduction [i] based on (23, 56, 514)-net in base 128, using
- base change [i] based on digital (16, 49, 514)-net over F₂₅₆, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 16, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [i]
    2. digital (0, 33, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(54−31, 54, 604)-Net over F₁₂₈ — Digital

Digital (23, 54, 604)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(54−31, 54, 1410704)-Net in Base 128 — Upper bound on s

There is no (23, 54, 1410705)-net in base 128, because

1 times m-reduction [i] would yield (23, 53, 1410705)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 4809 856923 806919 132162 080016 711802 722700 873619 478350 334606 062841 888362 855651 929454 660415 919000 911825 753653 063792 > 128⁵³ [i]